Understanding Various Types of Functions

Sep 12, 2024

Lecture on Functions

Overview

This lecture covers various types of functions and related concepts, including:

  • Relations and Functions
  • Inverse Functions
  • Composite Functions
  • Quadratic Functions
  • Domain and Range
  • Piecewise Functions
  • Even and Odd Functions
  • Polynomials
  • Rational, Modulus, and Radical Functions

Relations and Functions

  • Relation: A set of pairs of input (X values) and output (Y values).
    • X values represent the domain.
    • Y values represent the range.
  • Function: A specific type of relation where each input value has exactly one output value.
    • Examples:
      • 1,4 maps to 3,4 and 7,3 is a function.
      • Example with 1 having two outputs 1,2 and 1,3 is not a function.

Types of Functions

  • One-to-One Function: Each element in the domain maps to a unique element in the range.
  • Many-to-One Function: Multiple elements in the domain map to a single element in the range.
  • One-to-Many Function: One element in the domain maps to multiple elements in the range (not a function).
  • Many-to-Many Function: Does not qualify as a function.

Testing for Functions

  • A function must map each input to one output.
  • If two outputs map to the same input, it's not a function.
  • Example equations:
    • f(x) = x^2 - 1
    • Checking if f(a) = f(b) can determine if one-to-one.

Composite Functions

  • Combining two functions, e.g., f(g(x)).
  • Example: If f(x) = 2/x and g(x) = 1/(x+1), f(g(x)) involves substituting g(x) wherever x appears in f(x).

Inverse Functions

  • To find the inverse, swap x and y in the function and solve for x.
  • Example: If f(x) = 2/(x-2), find f^(-1)(x).

Domain and Range

  • Domain: All possible input values (X values).
  • Range: All possible output values (Y values).
  • Special considerations for functions with fractions or square roots.

Even and Odd Functions

  • Even Function: f(-x) = f(x).
  • Odd Function: f(-x) = -f(x).
  • Neither if neither condition is satisfied.

Quadratic Functions

  • Finding turning points using completing the square method.
  • Line of symmetry and intercepts.

Piecewise Functions

  • Functions defined by different expressions over different intervals.
  • Important to understand domain restrictions for each piece.

Graphing Functions

  • Important for visualizing domain, range, and intercepts.
  • Use turning points and symmetry to assist in sketching.

Polynomials, Rational, Modulus, and Radical Functions

  • Polynomials: Expressions involving powers of x.
  • Rational Functions: Ratios of polynomials.
  • Modulus Functions: Involve absolute values.
  • Radical Functions: Involve roots of expressions.

This lecture provides a comprehensive overview of functions, emphasizing understanding different types and how to manipulate them through algebraic operations.